Malcolm A. Sabin

Advances in Multiresolution for Geometric Modelling

Compression.- Recent Advances in Compression of 3D Meshes.- Shape Compression using Spherical Geometry Images.- Data Structures.- A Survey on Data Structures for Level-of-Detail Models.- An Algorithm for Decomposing Multi-dimensional Non-manifold Objects into Nearly Manifold Components.- Encoding Level-of-Detail Tetrahedral Meshes.- Multi-Scale Geographic Maps.- Modelling.- Constrained Multiresolution Geometric Modelling.- Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points.- Parameterization.- Surface Parameterization: a Tutorial and Survey.- Variations on Angle Based Flattening.- Subdivision.- Recent Progress in Subdivision: a Survey.- Optimising 3D Triangulations: Improving the Initial Triangulation for the Butterfly Subdivision Scheme.- Simple Computation of the Eigencomponents of a Subdivision Matrix in the Fourier Domain.- Subdivision as a Sequence of Sampled Cp Surfaces.- Reverse Subdivision.-

Polynomial reproduction by symmetric subdivision schemes

\sqrt 5

NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes

-subdivision.- Geometrically Controlled 4-Point Interpolatory Schemes.- Thinning.- Adaptive Thinning for Terrain Modelling and Image Compression.- Simplification of Topologically Complex Assemblies.- Topology Preserving Thinning of Vector Fields on Triangular Meshes.- Wavelets.- Periodic and Spline Multiresolution Analysis and the Lifting Scheme.- Nonstationary Sibling Wavelet Frames on Bounded Intervals: the Duality Relation.- Haar Wavelets on Spherical Triangulations.

A family of subdivision schemes with cubic precision

A novel 4-point ternary interpolatory subdivision scheme with a tension parameter is analyzed. It is shown that for a certain range of the tension parameter the resulting curve is C2. The role of the tension parameter is demonstrated by a few examples. There is a brief discussion of computational costs.

The detailed semantics of graphics input devices

We first present necessary and sufficient conditions for a linear, binary, uniform, and stationary subdivision scheme to have polynomial reproduction of degree d and thus approximation order d+1. Our conditions are partly algebraic and easy to check by considering the symbol of a subdivision scheme, but also relate to the parameterization of the scheme. After discussing some special properties that hold for symmetric schemes, we then use our conditions to derive the maximum degree of polynomial reproduction for two families of symmetric schemes, the family of pseudo-splines and a new family of dual pseudo-splines.

Eigenanalysis and Artifacts of Subdivision Curves and Surfaces

Regularity Criteria for the Topology of Algebraic Curves and Surfaces.- Quadrangle Surface Tiling Through Contouring.- Surfaces with Piecewise Linear Support Functions over Spherical Triangulations.- A Developable Surface of Uniformly Negative Internal Angle Deficit.- Rational Maximal Parametrisations of Dupin Cyclides.- Discrete Harmonic Functions from Local Coordinates.- Computing the Topology of an Arrangement of Quartics.- Non-uniform B-Spline Subdivision Using Refine and Smooth.- Scattered Data Fitting on Surfaces Using Projected Powell-Sabin Splines.- Implicit Boundary Control of Vector Field Based Shape Deformations.- Tuning Subdivision Algorithms Using Constrained Energy Optimization.- Description of Surfaces in Parallel Coordinates by Linked Planar Regions.- Discrete Surface Ricci Flow: Theory and Applications.- Guided C 2 Spline Surfaces with V-Shaped Tessellation.- MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries.- Mean Value Bezier Surfaces.- Curvature Estimation over Smooth Polygonal Meshes Using the Half Tube Formula.- Segmenting Periodic Reliefs on Triangle Meshes.- Estimation of End Curvatures from Planar Point Data.- Inversion, Degree and Reparametrization for Rational Surfaces.- Discrete Surfaces in Isotropic Geometry.- An Appropriate Geometric Invariant for the C 2-Analysis of Subdivision Surfaces.- Curvature-Based Surface Regeneration.- Bounded Curvature Subdivision Without Eigenanalysis.- Facial Shape-from-Shading Using Principal Geodesic Analysis and Robust Statistics.- Statistical Methods for Surface Integration.- Skeleton Surface Generation from Volumetric Models of Thin Plate Structures for Industrial Applications.- Parallel Tangency in .- Condition Numbers and Least Squares Regression.- Propagation of Geometric Tolerance Zones in 3D.

Recent Progress in Subdivision: a Survey

We present a subdivision framework that adds extraordinary vertices to NURBS of arbitrarily high degree. The surfaces can represent any odd degree NURBS patch exactly. Our rules handle non-uniform knot vectors, and are not restricted to midpoint knot insertion. In the absence of multiple knots at extraordinary points, the limit surfaces have bounded curvature.

Two-dimensional potential fields for advanced implicit modeling operators

The four-point subdivision scheme is well known as an interpolating subdivision scheme, but it has recently come to our notice that it is but the first scheme in a family all of whose members have the property that if all the control points lie equally spaced along the same cubic polynomial, the limit curve is exactly that polynomial. Other members of the family have higher smoothness. We study these schemes as functions, where the ordinate is given by the scheme, while the abscissae of the control points are equally spaced. Because all schemes include linear functions in their precision set, this may be regarded as a particular case of the parametric setting, rather than as a special case. This paper introduces the family and determines how the support, the Holder regularity, the precision set, the degree of polynomials spanned by the limit curves, and the artifact behavior vary with the integer parameter that identifies the members of the family. For the family members with an even parameter value, most of these properties have also been shown by Dong and Shen (Dong, B., Shen, Z., 2007. Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22 (1), 78-104), as they turn out to be a particular kind of pseudo-splines. But regarding the regularity exponent of the limit functions, we derive the exact values and thus improve the lower bounds given by Dong and Shen in that paper. Moreover, our analysis also covers the family members with an odd parameter value which do not seem to fit into the general framework of pseudo-splines. Just before this paper was submitted, (Choi, S.W., Lee, B.-G., Lee, Y.J., Yoon, J., 2006. Stationary subdivision schemes reproducing polynomials. Comput. Aided Geom. Design 23 (4), 351-360) appeared, which also discusses a family of subdivision schemes. The high order members of that family achieve higher degrees of polynomial reproduction, whereas ours aim only at cubic reproduction. This allows us to gain higher continuity for a given mask width.